I read about Helmholtz theorem on the Griffiths (Introduction to electrodynamics) and in the appendix B it shows how to use the Helmholtz theorem to determine the field F starting from the potential. The book says that:

Suppose we are told that the divergence of a vector function F(r) is a specified scalar function D(r):
$\nabla \cdot \textbf{F}=D(r)$

and the curl of F(r) is a specified vector function C(r):
$\nabla \times \textbf{F}=\textbf{C}$

For consistency, C must be divergenceless,
$\nabla \cdot \textbf{C}=0$
because the divergence of a curl is always zero. Question: can we, on the basis of this information, determine the function F? If D(r) and C(r) go to zero sufficiently rapidly at infinity, the answer is yes, as I will show by explicit construction.
I claim that